Analysis of dynamic models for turbulent premixed combustion

Center for Turbulence Research Proceedings of the Summer Program 2012 387 Analysis of dynamic models for turbulent premixed combustion By D. Veynante, T. Schmitt, M. Boileau AND V. Moureau Very few attempts have been made to implement dynamic combustion models in large eddy simulations, whereas they appear to be a promising alternative to flame surface density balance equations to handle situations where flame wrinkling and turbulence are not in equilibrium, as assumed in usual algebraic formulations. Dynamic formulations for the flame wrinkling factor are investigated here, from both a priori (DNS) and a posteriori tests. The emphasis is on: (i) the model s ability to handle laminar flame situations; (ii) the efficiency of the procedure s implementation, especially on unstructured meshes and massively parallel machines and (iii) the determination of local, time- and space-dependent model parameters. Dynamic modeling is found to be very powerful, with acceptable extra cost. 1. Introduction Large eddy simulations (LES) are now widely used to describe turbulent premixed combustion (Janicka & Sadiki 2005; Pitsch 2006; Poinsot & Veynante 2011). This technique gives access to unsteady flame behaviors as encountered during transient ignition (Boileau et al. 2008), combustion instabilities (Menon & Jou 1991) or cycle-to-cycle variations in internal combustion engines (Richard et al. 2007). The unresolved flame/turbulence interactions may be modeled in terms of subgrid scale turbulent flame speed (Pitsch 2006), flame surface density (Boger et al. 1998) or flame surface wrinkling factor (Colin et al. 2000; Charlette et al. 2002a). These models generally assume an equilibrium between turbulence motions and flame surface, expressed through algebraic expressions which are not adapted to transient situations (Richard et al. 2007). Some authors suggest to solve an additional balance equation for the flame surface density (Hawkes & Cant 2000; Richard et al. 2007) or the flame wrinkling factor (Weller et al. 1998). Dynamic models that take advantage of the known resolved flow field to automatically adjust model parameters during the simulation are a priori able to handle situations where an equilibrium has not yet been reached between turbulence motions and flame movements. However, while this approach has been used routinely for unresolved transport since the pioneering work of Germano et al. (1991), relatively few attempts have been made to develop dynamic combustion models (Charlette et al. 2002b; Knikker et al. 2004; Pitsch 2006; Knudsen & Pitsch 2008; Poinsot & Veynante 2011). Recently, Wang et al. (2011, 2012) have shown the ability of a dynamic flame wrinkling factor model to reproduce a statistically steady jet flame (Chen et al. 1996) and the transient ignition of a flame kernel (Renou et al. 2000) under several operating conditions, a unique parameter that only evolves with time being determined by the dynamic procedure. The objective of this project is to go further in the analysis, formulation and practical implementation of flame wrinkling factor dynamic models. This wrinkling factor is EM2C, CNRS UPR 288, Ecole Centrale Paris, France CORIA, CNRS UMR 6614, Université et INSA de Rouen, Saint-Etienne du Rouvray, France

388 Veynante et al. Model W (ec) Boger et al. (1998) 4ρ us l (6/π) 1/2 ec (1 ec) Level set / G-equation ρ us ` 2 l + δl 2 1/2 G TFLES δ l ω (ec) F-TACLES ` 2 + δl 2 1/2 F l (ec, ) Table 1. Generic reaction rate expressions according to Eq. (2.1) for four turbulent premixed combustion LES models. ρ u is the fresh gases density, S l and δ l the laminar flame speed and thickness, respectively. G is the level-set field, usually defined as the signed distance to the flame front. F l (ec, ) is given by filtering one-dimensional laminar premixed flames (Fiorina et al. 2010). a basic ingredient to describe interactions between flame fronts and turbulence motions in combustion models such as the level-set (Pitsch 2006; Knudsen & Pitsch 2008), the thickened flame model (Colin et al. 2000; Charlette et al. 2002a), algebraic flame surface density models (Boger et al. 1998), or the recently developed F-TACLES approach (Fiorina et al. 2009) to incorporate complex chemistry features in LES. Our attention focuses on: (i) the correct propagation of laminar premixed flames, observed during the early stages of flame development; (ii) the use of a local model parameter, Wang et al. (2011, 2012) computations considering only a time-evolving spatially uniform parameter; and (iii) the practical implementation of dynamic models. Our approach combines theoretical analysis (Section 2), DNS a priori tests (Section 3) of the lean premixed swirled PRECCINSTA turbulent flame (see Moureau et al. 2010, 2011 for details), and preliminary LES (a posteriori tests) discussed in Section 4. 2. Theory and modeling 2.1. Generic formulation of the reaction rate The filtered reaction rate to be modeled in LES is written under the generic form W ( c) ω (c) = Ξ 2 + δ, (2.1) 2 l where c is the reaction progress variable, increasing from c = 0 in fresh gases to c = 1 in fully burnt products, but here stands for any quantity entering the reaction rate. W ( c)/ 2 + δl 2 corresponds to the resolved reaction rate, estimated from filtered quantities such as the mass-weighted filtered progress variable c. is the LES filter size. The wrinkling factor Ξ measures the ratio of total to resolved flame surfaces in the filtering volume. The laminar flame thickness δ l is introduced here in the Charlette et al. (2002b) expression to ensure correct behavior when 0. Equation (2.1) holds as long as flame/turbulence interactions are described in terms of flame surface wrinkling factor or subgrid scale turbulent flame speed S T = Ξ S l, where S l is the laminar flame speed (flamelet assumption). Table 1 summarizes W expressions for the Boger et al. (1998) algebraic model, the level-set approach (Pitsch 2006), the thickened flame model (Colin et al. 2000) where the thickening factor is given in terms of flame thickness and filter size as F = [( /δ l ) 2 + 1] 1/2 and the F-TACLES model based on the filtering of one-dimensional laminar premixed flames (Fiorina et al. 2010).

Dynamic models for LES of turbulent combustion 389 Figure 1. Fresh gas subgrid scale turbulence intensity, u (m/s, left) and wrinkling factor Ξ (right) as a function of / x. Bold line: DNS; circles: Charlette et al. (2002a) expression (Eq. 2.2, β = 0.337); thin lines: saturated Charlette et al. (2002a) expression (Ξ = ( /δ l ) 0.337 ). is the LES filter size, x the mesh size, δ l = 3.4 x. Only a small part of the DNS (cubic box of 6 6 6 mm 3 volume, centered on location x 0 1.4, y 0 0.32, z 0 1.9, in cm) is considered. Charlette et al. (2002a) model the wrinkling factor Ξ by the algebraic expression ( [ ( ) ( ) ]) Ξ = 1 + min max 1, 0, Γ, u u β, Re, (2.2) δ l δ l S l where the efficiency function Γ describes the ability of vortices to effectively wrinkle the flame front, β is the model parameter, u and Re = u /ν the subgrid scale turbulence intensity and Reynolds number, respectively, with ν representing the fresh gas kinematic viscosity. Following Wang et al. (2012), the original expression has been modified to maximize Ξ by ( /δ l ) β for large turbulence intensities ( > δ l ). The model parameter is then related to the fractal dimension D of the flame surface through β = D 2 (Gouldin 1987; Gouldin et al. 1989; Gulder 1991). Equation (2.2) assumes an equilibrium between turbulence motions and flame surface (Colin et al. 2000), which is not reached during early flame developments, when the flame is initially laminar and progressively wrinkled by turbulence motions, for example close to injector lips or following a spark ignition. These situations could be handled by solving an additional balance equation relaxing the equilibrium assumption. An alternative is to automatically adjust the model parameter from the known resolved flame motions (dynamic formalism). This work investigates in detail this second solution. 2.2. First DNS analysis The following modeling derivation is supported by a first DNS analysis. Figure 1 displays the subgrid scale turbulence intensity in fresh gases, u, and the wrinkling factor Ξ as a function of the ratio between filter-size and mesh-size x, together with wrinkling factor modeling. The fresh gas subgrid scale turbulence intensity is estimated as ) (u u V 2 c u (uc u ) 2 1/2 /c u dv = V c, (2.3) u dv where u is the velocity vector, c u the conditioning variable (c u = 1 for c 0.04, c u = 0 elsewhere) and V the box volume, denoting the filtering operation. The threshold value is set as a compromise between the number of available fresh gas samples and the relevance S l

390 Veynante et al. of the estimation. As expected, u increases with the filter size to reach about 4 m/s for / x = 20, corresponding to u /S l 13.6 (S l = 0.293 m/s). The wrinkling factor Ξ increase with is well-recovered by Eq. (2.2), noting that points below / x = 5 have little physical significance, as becomes of the order of the laminar flame thickness δ l. Figure 1 also indicates that Eq. (2.2) is saturated (i.e., the minimum term is given by the ratio /δ l and independent of the turbulence intensity u ), as noted in most practical implementations, reducing to ( ) β Ξ = (2.4) The best fitting value, β = 0.337, corresponds to a fractal dimension D = β+2 = 2.337, in agreement with Gouldin (1987) and very close to D = 7/3 (Kerstein 1988). However, a constant fractal dimension would correspond to a uniform wrinkling factor over all the flow field, which is generally not verified. Equation (2.4) with local- and time-dependent β values is more general than a usual fractal model. δ l 2.3. Dynamic formalism The parameter β is determined by equating the reaction rate averaged over a given volume (< >), which is evaluated at the LES-filter ( ) and test-filter ( ) scales: ( ) ) β (γ ) β W ( c) W γ ( c =, (2.5) δ l 2 2 + δ l δ l (γ ) 2 2 + δ l where c denotes a mass-weighted filtering at scale of the filtered progress variable c. γ = ( 2 + 2 ) 1/2 is the effective filter size when combining LES and test Gaussian filters. Equation (2.5) provides a relation to evaluate β, assuming that the wrinkling factor is uniform over the averaging volume. We focus here on two key requirements: (a) To correctly recover unity wrinkling factors (Ξ = 1), i.e., β = 0, when the wrinkling of the flame front is fully resolved in simulations. (b) To replace the averaging operation <. > by a Gaussian filter, which is easier to implement for unstructured meshes and/or on massively parallel machines (diffusion operation). The first requirement supposes that laminar flames verify Eq. (2.5) with β = 0, which is generally not true as W -expressions only approximate the actual reaction rates. Wang et al. (2011) enforced this condition through a calibration factor in the thickened flame model. The second condition requires that W ( c) and W γ ( c) are similar for planar laminar flames to avoid unforeseen bias due to the Gaussian filtering replacing the averaging procedure. The best solution found is to recast Eq. (2.5) in terms of flame surfaces: Ξ ĉ c = Ξ γ, (2.6) where c, Ξ c, ĉ and Ξ γ ĉ measure resolved and total flame surface densities (i.e. flame surface per unit volume) at LES and test-filter scales, respectively. Assuming that β is uniform over the averaging volume and independent of the filtering scale, combining Eqs. (2.4) and (2.6) leads to ( ) ln c ĉ / β = (2.7) ln (γ)

Dynamic models for LES of turbulent combustion 391 Figure 2. Distribution of the model parameter β (numbers of samples, left); modeled reaction rate according to Eqs. (2.1) and (2.4) combined with the TFLES model (Table 1) as a function of the filtered reaction rate ω(c) extracted from DNS (right). / x = 6, b / x = 9, m/ x = 20. 95306 samples are displayed from the same DNS part as in Figure 1. For laminar planar flames, c = ĉ and β = 0, fitting the two requirements above. Unfortunately, Eq. (2.7) involves only filtered quantities instead of Favre-filtered quantities that are solved for in LES. However, for infinitely thin flame fronts, filtered progress variables are linked through ρ c = ρc = ρ b c ; ρ c = ρc = ρ b ĉ, (2.8) where ρ b is the burnt gas density. These relations suggest the approximations ( ) ( ) ln ρ c / ρ c ln c / c β, (2.9) ln (γ) ln (γ) which have been confirmed from DNS-analysis. Averaging (or Gaussian filtering) < > acts as an integration across the resolved flame front, measuring the filtered progress variable step between fresh and burnt gases. Equation (2.9) implicitly assumes that the size of this Gaussian filter, m, is larger than the resolved flame front (typically, in the following tests, 1.5 m 3, where is the test-filter scale). 3. A priori tests Analyses are first conducted on a reduced part of the DNS database as in Section 2.2. 3.1. Reduced database Figure 2 compares the filtered reaction rate modeled by combining the TFLES model (Table 1) with Eqs. (2.1) and (2.4) to the reaction rate ω(c) extracted from DNS. The agreement is very good (correlation coefficient of about 0.99) even though the model slightly underestimates (respectively, overestimates) low (large) reaction rate values. The corresponding distribution of the model parameter is also displayed, this confirms that β cannot be assumed uniform and provides further justification for a dynamic formalism. 3.2. Full database (2.6 billion cells) Figure 3 displays an instantaneous β field that was extracted from the DNS using the first approximation in Eq. (2.9). As expected, the model parameter starts from low values and increases as the flame is progressively wrinkled by turbulence and convected downstream. Large β values close to unity are observed at the flame tip when burning pockets detach from the main flame. Note that β evolves smoothly and would be compatible with

392 Veynante et al. Figure 3. Side (left) and upstream (right) instantaneous view of c = 0.8 isosurface colored by b = 1.5, m = 3. the model parameter β. Filter sizes: = 10 x, Figure 4. Left: evolution of β with the downstream location (cm). Bold line: mean (βm ); dashed line: rms; thin line: β estimated using transverse slices of 1 mm thickness as averaging volume; dotted-dashed line: global β value (averaging over the computational domain). Right: total (St, bold line), resolved (Sr, thin line) and modeled (Sm, dots) flame surfaces, defined by Eqs (3.1 3.2) and given in cm2, as a function of the downstream location (cm). Averages are computed over transversal slices of 1 mm length along the downstream direction. Filter sizes identical to Figure 3. The flame thickness entering the wrinkling factor (Eq. 3.2) is set to δl = 3.4 x. numerical simulations. These findings are confirmed in Figure 4 displaying the evolution of mean and rms β values with the downstream direction. The mean value progressively increases from β 0.1 at the burner inlet up to β 0.6 at x = 2 cm. Then, β is roughly constant for 2 x 4 cm, denoting turbulence/flame equilibrium. In the last phase (4 x 5.5 cm), β strongly increases which can be associated with the formation of flame pockets. The rms is roughly constant and about 0.2. The 1-D β-values, computed using transverse slices of 1 mm thickness in the downstream direction as averaging volume h i, are very similar up to x 4 cm. The discrepancies observed downstream are due to flame pockets: mean β-values are conditioned on the flame front while 1-D β are smoothed by the averaging volume, and correspond to very low flame surfaces (see Figure 4, right). The global parameter value, which is evaluated by volume-averaging over the computational domain, is also indicated (β 0.51) and overestimates the front wrinkling factors during the development phase of the flame (x 2 cm). Figure 4 (right) displays the downstream evolution of the total (St ), resolved (Sr ) and

Dynamic models for LES of turbulent combustion 393 Figure 5. Instantaneous field of the filtered progress variable ec colored by the model parameter value in the LES of the Chen et al. (1996) jet flame (Case F3, inlet jet velocity 30 m s 1, stoichiometric conditions). COLOR modeled (S m ) flame surfaces estimated as S t (x) = + + x+ x x c dx dy dz ; S r (x) = ( S m (x) = δ l + + x+ x x c dx dy dz (3.1) ) βm(x) S r (x), (3.2) where x = 1 mm and β m (x) is the mean value displayed in Figure 4. The resolved flame surface is lower than the total flame surface by about 30%, while the total flame surface is very well estimated from Eq. (3.2), validating the dynamic fractal-like model. 4. A posteriori tests The proposed formalism is implemented in the structured low-mach code FASTEST from TU-Darmstadt (Germany) to perform LES of the Chen et al. (1996) premixed methane/air F3 jet flame, stabilized by a coflow of burnt gases, for which results are already available for time-evolving spatially uniform β values (Wang et al. 2011). The dynamic procedure is combined with F-TACLES as done in Schmitt et al. (2013). The mesh contains 2,800,000 hexahedra and grid spacing is kept constant over the region of interest ( x =0.4 mm, while the injector diameter is d=12 mm). The filter size for the F-TACLES model and the test-filter size are set to 2 mm and 3 mm, respectively. The averaging Gaussian filter size m is 7 mm. The cut-off length scale δ l in Eq. 2.2 is set to 2/ max( c ), estimated for the unfiltered planar laminar flame. Simulation is averaged over 4 convective times τ c (τ c = L fl /u 0 = 5 ms, where L fl = 0.15 m is the flame length and u 0 = 30 m/s is the inlet bulk velocity), using 1400 CPU hours on a cluster Altix ICE 8400. Figure 5 displays a snapshot of the turbulent jet flame where an iso-surface of the progress variable is colored by the local model parameter value. As expected, β values are small but increase in the initial flame region downstream of the injector lip, as the flame is progressively wrinkled by turbulence motions. Then, β reaches a plateau of about β = 0.5 (the value recommended by Charlette et al. 2002a) when flame surface wrinkling and turbulence motions are in equilibrium. Large values are observed in the flame tip. This evolution is confirmed by Figure 6 (top left) displaying the downstream evolution of the mean β value. Figure 6 compares mean methane and carbon dioxide mass fraction profiles with experimental data from Chen et al. (1996). The agreement is very good but analysis of simulations will be refined in the near future, with particular detailed emphasis on the unsteady behavior of the flame with uniform and local dynamic parameters.

394 Veynante et al. 1 15 [-] Y [%] 0.8 0.6 0.4 0.2 0 0 2 4 6 8 x/d [-] 10 12 14 15 10 x=6.5d Y CO2 5 Y CH4 0 0 0.5 1 1.5 r/d [-] Y [%] Y [%] 10 x=4.5d Y CO2 5 Y CH4 0 0 0.5 1 1.5 r/d [-] 15 10 x=8.5d Y CO2 5 Y CH4 0 0 0.5 1 1.5 r/d [-] Figure 6. Top left: downstream evolution of the β conditional average (for 0.6< c <0.8) in the Chen et al. (1996) turbulent jet flame. Downstream coordinate x is made non-dimensional by the jet diameter d=12 mm. For a given axial position x, the average is performed in a domain of thickness 1 mm centered on the x location. The averaging time is 0.4 τ c. Top right and bottom: transverse filtered methane e Y CH4 and carbon dioxide e Y CO2 mass fraction profiles for three downstream locations x/d = 4.5, 6.5 and 8.5. Symbols: experiments (Chen et al. 1996); lines: simulation. 5. Conclusions A flame wrinkling factor dynamic formalism was developed and investigated for large eddy simulations of turbulent premixed combustion from theoretical analysis, a priori tests processing a DNS database of a turbulent swirled flame and preliminary a posteriori tests simulating a turbulent jet flame. Flame wrinkling factors, measuring the ratio of total to resolved flame surfaces in the filtering volume, enter directly, or indirectly through subgrid scale turbulent flame speed, various combustion models. They are generally modeled through algebraic expressions that assume an equilibrium between turbulence motions and flame dynamics, which is generally not reached at early stages of flame developments, for example close to the injector lips in steady-state configurations or after spark ignition where the flame is usually laminar and progressively wrinkled by turbulence motions when growing or convected downstream. Dynamic models appear as a promising alternative to balance equations to handle non-equilibrium situations, as shown by the results presented. Attention was focused on three key points: (i) the ability to correctly predict the propagation of a laminar flame front that could be encountered in some locations in the flow field because of the refined meshes now available; (ii) replacement of the averaging volume, required to determine resolved and test-filtered flame wrinkling, by a Gaussian operator easier to implement on unstructured meshes and/or massively parallel machines (diffusion operator); (iii) the use of a local model parameter, evolving both in space and time. The two first requirements suggest to propose a formalism based on conservation of the flame surface instead of on chemical reaction rates. Both a priori and a posteriori

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